Yeah at this point ... this is probably not a good idea.
Further: That's a casing from a pistol round, probably 9mm, not a necked casing from a hunting rifle.
Apparently, the entire 'trans ideology on the casings' thing ... came from our favorite divorced domestic abuser Steven Crowder, who claims that basically just someone in the ATF just sent either their own opinion or a prelim. analysis to him.
Then, as best I can tell... the WSJ just ran with that, and then everyone else did.
Then the NYT appears to have actually attempted to do journalism, reached out to the ATF for comment/verification, and they basically said 'wtf no, that document is preliminary at best and should not be seen as accurate'... which to me implies they couldn't even be 100% certain as to what specific document Crowder / WSJ was even referring to.
So... hooray journalism I guess, basically everything is a wildly irresponsible gossip rag right now.
EDIT:
Handy dandy idiots guide to bullet cartridge calibers and casings, if anyone cares.

Your modern rifle rounds are generally gonna be necked like the .223/5.56, generally taller, generally larger bullets.

bullet = part that hits thing you are aiming at, ideally.
casing = part that holds gunpowder, has primer, gets hit by firing pin, must be cleared from the chamber before next shot.
cartridge/round = the whole package, all together.
There are still some non necked, straight cased rounds like the 30 30, still somewhat common in lever action ("cowboy repeater") type rifles/carbines.
There are also some smaller rounds that are necked, basically tiny, scaled down rifle style rounds, that are geared toward varmint hunting, or a personal defense context, via moving a smaller and lighter bullet faster, with relatively more gunpowder behind it.... used variously in more niche pistols, rifles, pdws, carbines.
5.7 x 28mm (the 5-7 / P90), 4.6 x 30mm (the MP7), .17 HMR (decent array of varmint rifles), etc.